The subject matter of the present invention relates to a Digital Pressure Derivative software, adapted to be stored in a computer system, for receiving a measurement function including ‘noise’, such as a measurement of pressure in an oil or gas well, for convolving the measurement function with a wavelet, and for generating a high resolution and high accuracy output signal which is substantially devoid of the ‘noise’ and which will enable a customer to interpret the high resolution and high accuracy output signal in order to obtain more information about the measurement. When the measurement function represents a measurement of pressure in an oil or gas reservoir, the pressure measurement function including the ‘noise’ is convolved with a derivative wavelet thereby generating a much higher resolution ‘derivative of the pressure measurement function’ which is substantially devoid of the ‘noise’, the absence of the ‘noise’ in the higher resolution ‘derivative of the pressure measurement function’ generated by the convolution enabling a customer to interpret the ‘derivative of the pressure measurement function’ and obtain more information about the pressure measurement and the reservoir.
Reservoir characterization using well test data has made remarkable progress in the last decade. Significant advances in mathematical techniques have permitted the analytical solution to complex reservoir flow problems which were previously obtainable only by means of numerical methods. Notable contributions in this area include the work of Ozkan and Raghavan1-3 in which the Laplace transform and the method of sources and sinks were systematically combined to compute pressure distributions for various well and reservoir complexities. Utilizing the Laplace transform allows the addition of natural fractures,4 variable rate production5 (wellbore storage) and commingled production6 in a general and efficient manner. Combined with the Stehfest7 algorithm for numerically inverting the Laplace space solutions into the time domain we have, from a software implementation standpoint, a straightforward approach to developing efficient and extendible libraries of analytical solutions to the diffusivity equation. For situations in which analytical procedures are limited (e.g., complex geometries, multiphase flow) we may use recently developed software specifically designed to interactively model transient welltest data using finite-difference methods (e.g., GeoQuest's WellTest 200).
The advances made in the measurement of pressure transient data have complimented these new solution techniques by providing the stability and resolution required to perceive complexities of the well/reservoir system which may otherwise have gone unnoticed. Currently, we are able to measure pressures with a resolution of 0.001 psi with 10 second sampling and with a stability (drift) of less than three psi/year.8 
The interpretation and measurement of well test data have, with a few recent exceptions, advanced separately. Notable exceptions include two studies presented by Veneruso et al.9,10 The first demonstrates how errors in pressure measurement affect well test analysis using a transfer function approach. Gauge effects, such as drift and resolution are incorporated into the analytical simulation of well tests in order to optimize testing performance with cost-effective instruments. The second discusses key data acquisition and data transmission techniques in addition to giving a good overview of the electronic processes involved in making pressure, flowrate and temperature measurements.
In this specification, we hope to further reduce the separation between advances in well test interpretation and advances in measurement technology by incorporating our knowledge of one within the other. Specifically, we utilize the firm mathematical foundation of Digital Signal Processing to combine the characteristics of the well test measurement process with our knowledge of reservoir physics to improve the pressure derivative technique of diagnosing and interpreting pressure transient data. The term derivative as used herein is intended to imply differentiation with respect to the time function plotted on the abscissa of the appropriate semi-log plot; for generality, this should be considered the superposition time function.
Arguably the most significant advance in well test interpretation the last decade, the pressure derivative, as described by Bourdet et al.,11-14 has become the primary tool for diagnosing well and reservoir behavior. It provides the basis for modern well test interpretation methodology15 and has become a customary and requisite feature in the myriad commercial well test interpretation software now available. In many situations however, the derivative of the measured data is uninterpretable or, worse, mis-interpreted by the analyst because of various artifacts of the measuring and differentiating process collectively termed “noise”. A published article16 states “The main disadvantage of the pressure derivative is that one must construct derivative data by numerical differentiation of measured pressure data. The resulting pressure derivative data often are noisy and difficult to interpret. While various “smoothing” techniques have been used to reduce noise, some concern exists that the smoothing procedures may alter the basic character of the data.” This opinion is respected by most engineers involved with the interpretation of well test data.
Various algorithms have been used in an effort to eliminate or reduce the inevitable noise associated with the numerical differentiation of measured data. For example, Bourdet, et al.14 considers polynomial fitting of the data and taking the analytical derivative of the polynomial. This procedure, in addition to being cumbersome, alters the shape of the original data. A second procedure given is to compute the third derivative of the data and subsequently integrating twice. This method tends to give false oscillations at late times and during infinite acting radial flow. The current practice is to use centuries-old techniques developed by Newton and Stirling for interpolating data tables (backward, forward, or central difference typically utilizing three points). When viewed in the frequency domain, these techniques are clearly highpass and bandpass filters, which exaggerate high frequency noise and distort the “true” dp/dt curve. The data is then typically smoothed by choosing the points used in the calculation a sufficient distance from the point of interest. This distance, given as L in Ref. 14, is expressed in terms of the appropriate time function. The idea of smoothing the derivative is considered suspect by some due to the subjective choosing of L. If L is chosen too large, the character of the actual signal will be distorted. Nevertheless, judging by the standard use of this method in commercial well testing software, it apparently has become the most popular. Akima17 suggests a slightly different approach whereby the slope of the point in question and two points on each side are used. A third degree polynomial representing a portion of the curve between a pair of given points is determined by the coordinates and the slopes at the two points. The derivative of the polynomial is then computed.
Regardless of which method of numerical differentiation is used and regardless of the chicanery employed to reduce the scatter in the resulting derivative data, the analyst is often left with data not entirely representative of the well/reservoir system (s)he is trying to identify.
In this specification, we describe a new technique for differentiating well test pressure data called the Digital Pressure Derivative Technique (DPDT). The DPDT produces the most accurate and representative dp/dt curve by incorporating knowledge of both reservoir and gauge physics. It is efficient and straightforward to implement and may be used on either real-time or recorded data. No modifications to gauges or surface hardware are necessary. Furthermore, error bounds and a quality control mechanism, indicating the end of a transient (i.e. a signal-to-noise ratio test) can be provided allowing the optimal use of recorder memory space. Additional DSP techniques such as oversampling may be utilized to further enhance the quality of the DPDT. The DPDT is generated by convolving the measured pressure samples with a special wavelet. This wavelet is made from the unit sample response of a bandlimited, optimum linear phase finite impulse response (FIR) differentiator. The DPDT is based on the firm mathematical foundation of Digital Signal Processing (DSP), reservoir physics and the characteristics of the pressure measurement system.